Factorial moment

In probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments are useful for studying non-negative integer-valued random variables, and arise in the use of probability-generating functions to derive the moments of discrete random variables.

Factorial moments serve as analytic tools in the mathematical field of combinatorics, which is the study of discrete mathematical structures.

Definition
For a natural number $r$, the $r$-th factorial moment of a probability distribution on the real or complex numbers, or, in other words, a random variable $X$ with that probability distribution, is


 * $$\operatorname{E}\bigl[(X)_r\bigr] = \operatorname{E}\bigl[ X(X-1)(X-2)\cdots(X-r+1)\bigr],$$

where the $E$ is the expectation (operator) and


 * $$(x)_r := \underbrace{x(x-1)(x-2)\cdots(x-r+1)}_{r \text{ factors}} \equiv \frac{x!}{(x-r)!}$$

is the falling factorial, which gives rise to the name, although the notation $(x)_{r}$ varies depending on the mathematical field. Of course, the definition requires that the expectation is meaningful, which is the case if $(x)_{r}$ or $x(x - 1)(x - 2) ... (x - r + 1)$.

If $(X)_{r} ≥ 0$ is the number of successes in $E [$ trials, and $X$ is the probability that any $n$ of the $p_{r}$ trials are all successes, then


 * $$\operatorname{E}\bigl[(X)_r\bigr] = n(n-1)(n-2)\cdots(n-r+1)p_r$$

Poisson distribution
If a random variable $r$ has a Poisson distribution with parameter λ, then the factorial moments of $n$ are


 * $$\operatorname{E}\bigl[(X)_r\bigr] =\lambda^r,$$

which are simple in form compared to its moments, which involve Stirling numbers of the second kind.

Binomial distribution
If a random variable $X$ has a binomial distribution with success probability $X$$[0,1]$ and number of trials $X$, then the factorial moments of $p ∈$ are


 * $$\operatorname{E}\bigl[(X)_r\bigr] = \binom{n}{r} p^r r! = (n)_r p^r,$$

where by convention, $$\textstyle{\binom{n}{r}} $$ and $$(n)_r$$ are understood to be zero if r > n.

Hypergeometric distribution
If a random variable $n$ has a hypergeometric distribution with population size $X$, number of success states $X$ in the population, and draws $N$, then the factorial moments of $K ∈ {0,...,N}$ are


 * $$\operatorname{E}\bigl[(X)_r\bigr] = \frac{\binom{K}{r}\binom{n}{r}r!}{\binom{N}{r}} = \frac{(K)_r (n)_r}{(N)_r}. $$

Beta-binomial distribution
If a random variable $n ∈ {0,...,N}$ has a beta-binomial distribution with parameters $X$, $X$, and number of trials $α > 0$, then the factorial moments of $β > 0$ are


 * $$\operatorname{E}\bigl[(X)_r\bigr] = \binom{n}{r}\frac{B(\alpha+r,\beta)r!}{B(\alpha,\beta)} =

(n)_r \frac{B(\alpha+r,\beta)}{B(\alpha,\beta)} $$

Calculation of moments
The rth raw moment of a random variable X can be expressed in terms of its factorial moments by the formula


 * $$\operatorname{E}[X^r] = \sum_{j=0}^r \left\{ {r \atop j} \right\} \operatorname{E}[(X)_j], $$

where the curly braces denote Stirling numbers of the second kind.